Neutron Stars

A neutron star is what is left when the iron core of a massive star runs out of ways to hold itself up. Once electron degeneracy pressure fails at the effective Chandrasekhar mass, electron capture ( $p+e^-\to n+\nu_e$ ) and iron photodisintegration strip away support, and the core implodes in milliseconds until it reaches supranuclear density, where the strong interaction stiffens abruptly and the collapse rebounds. That bounce, reheated by neutrinos, powers a core-collapse supernova and leaves behind a city-sized ball of nuclear matter — or, for the heaviest progenitors, a black hole. Three facts organize everything that follows. First, the mean density is $\sim 10^{14}\text{–}10^{15}\,\mathrm{g\,cm^{-3}}$ — the density of an atomic nucleus, smeared over a 20 km sphere — so the microphysics is nuclear, not atomic. Second, the compactness $GM/Rc^2\approx0.2$ is of order unity, so Newtonian gravity is quantitatively wrong and the structure must come from the general-relativistic Tolman–Oppenheimer–Volkoff (TOV) equation. Third, integrating TOV for a chosen equation of state (EOS) yields a mass–radius curve whose peak is the maximum mass $M_{\max}$ . Because different nuclear models predict different peaks, and we now measure masses near $2\,M_\odot$ , radii near $12\,\mathrm{km}$ , and tidal deformabilities from mergers, neutron stars have become precision laboratories for matter at densities no accelerator can reach.

Key takeaways

Neutron stars are the compact remnants of massive stars ( $M\gtrsim8\,M_\odot$ ) whose iron cores collapse in supernovae, reaching and exceeding the nuclear saturation density $\rho_0\approx2.7\times10^{14}\,\mathrm{g\,cm^{-3}}$ .
They are held up by neutron degeneracy pressure augmented decisively by the repulsive core of the nuclear force; degeneracy alone gives only the Oppenheimer–Volkoff limit of $\sim0.7\,M_\odot$ , far below what is observed.
Canonical numbers: mass $\sim1.4\,M_\odot$ , radius $\sim10\text{–}13\,\mathrm{km}$ , mean density $\sim7\times10^{14}\,\mathrm{g\,cm^{-3}}$ , compactness $GM/Rc^2\approx0.21$ — so general relativity is mandatory.
Equilibrium obeys the Tolman–Oppenheimer–Volkoff equation, not Newtonian hydrostatics; solved for a given EOS it yields a maximum mass around $2.0\text{–}2.3\,M_\odot$ and a definite mass–radius relation.
Observed as radio pulsars, magnetars, X-ray binaries, and the merger GW170817, they probe matter above nuclear density; PSR J0740+6620 at $\approx2.08\,M_\odot$ and NICER radii near $12\,\mathrm{km}$ tightly bound the EOS.
Key idea: a neutron star is a general-relativistic equilibrium in which degeneracy pressure plus nuclear repulsion balances gravity; because its compactness is of order one, its mass–radius relation — computed from the TOV equation for a given dense-matter equation of state — directly encodes the physics of matter above nuclear density.

The core insight

A neutron star is a general-relativistic equilibrium in which degeneracy pressure plus nuclear repulsion balances gravity; because its compactness is of order unity, its mass–radius relation — computed from the TOV equation for a given dense-matter equation of state — directly encodes the physics of matter above nuclear density.

Where this lives today

The mass–radius plane is now being carved by measurements from three independent directions, and the surviving EOS is what remains inside all of them. Pulsar timing supplies the sharpest masses: the Shapiro delay in PSR J0740+6620 gives $M\approx2.08\,M_\odot$ , which alone excludes any EOS whose TOV peak falls below that line — including many models with large hyperon or condensate content. NASA's NICER X-ray telescope fits the rotating hot-spot pulse profiles of nearby pulsars (PSR J0030+0451, PSR J0740+6620) to extract radii, placing the canonical $1.4\,M_\odot$ star near $R\approx12\,\mathrm{km}$ . And GW170817 measured the tidal deformability imprinted on the inspiral, $\Lambda_{1.4}\lesssim800$ , cutting off the stiffest equations from the other side, while its kilonova AT2017gfo confirmed mergers as r-process factories forging gold and platinum. Open questions remain: does a hyperon or quark-matter core appear at the highest densities (the "hyperon puzzle"), what is the true maximum mass, and what was the $2.6\,M_\odot$ object in GW190814 — heaviest neutron star or lightest black hole? Next-generation gravitational-wave observatories (Einstein Telescope, Cosmic Explorer) and the Square Kilometre Array are built to answer them.

The physical picture

Picture squeezing the Sun until electrons are forced onto protons, the two combine into neutrons, and the whole star becomes a single gigantic nucleus held together not by chemical bonds but by gravity, with the Pauli exclusion principle and the nuclear force's short-range repulsion pushing back, in a sphere roughly 20 km across. Inside, the density climbs by eight orders of magnitude from a wispy ion atmosphere to several times nuclear density at the centre, each regime a different phase of matter. The first figure is that onion.

Figure

Interior of a canonical 1.4 M☉ neutron star: four layers from a thin ion atmosphere down to a supranuclear core, with density climbing from about 10⁶ g cm⁻³ at the surface to several times the nuclear saturation value ρ₀ ≈ 2.7×10¹⁴ g cm⁻³ at the centre, all inside a radius of only about 12 km

The reason a neutron star can outweigh a white dwarf, though both lean on degeneracy, is the nuclear force. An ideal neutron gas in GR (Oppenheimer–Volkoff, 1939) tops out at only $\sim0.7\,M_\odot$ ; it is the stiffness of the nuclear interaction, resisting compression far more than an ideal Fermi gas, that raises the ceiling to $\sim2\,M_\odot$ and beyond. Every EOS thus predicts its own peak mass and mass–radius curve — and the second figure is what the whole field reaches for: those curves, and the measurements that cut them down.

Figure

Mass–radius relation from integrating the TOV equation for two dense-matter equations of state: a stiff EOS reaches larger radii and a higher maximum mass near 2.3 M☉ while a soft EOS gives smaller radii and a lower peak near 2.0 M☉; the measured 2.08 M☉ pulsar PSR J0740+6620 excludes any EOS whose peak falls below the horizontal line, and NICER places the canonical 1.4 M☉ star near 12 km

General relativity is not a small effect here: the surface escape speed is a large fraction of $c$ , gravitational time runs measurably slow, and light bends enough to see over the limb onto the far hemisphere. Because the pressure at a given density is fixed by nuclear physics, every star of a given mass has (for the true EOS) a definite radius — so measuring masses and radii reverse-engineers matter we can never make in a laboratory.

The core equations

Start with the crudest diagnostic: packing $1.4\,M_\odot$ into a 10 km sphere gives a mean density that already announces the physics is nuclear, not atomic.

\bar{\rho} = \frac{3M}{4\pi R^3} \approx 6.7\times 10^{14}\ \mathrm{g\,cm^{-3}}\ \left(\frac{M}{1.4\,M_\odot}\right)\left(\frac{10\,\mathrm{km}}{R}\right)^3 \qquad (1)

where: $M$ — gravitational mass; $R$ — circumferential radius; $\bar\rho$ — mean mass density. Reading: the mean density is a few times the nuclear saturation density $\rho_0\approx2.7\times10^{14}\,\mathrm{g\,cm^{-3}}$ , with the centre higher still — matter held above nuclear density by gravity alone. The second diagnostic decides whether Newton is allowed. The compactness $GM/Rc^2$ is the ratio of the gravitational radius to the actual radius — how curved the surrounding spacetime is.

\frac{GM}{Rc^2} \approx 0.21\ \left(\frac{M}{1.4\,M_\odot}\right)\left(\frac{10\,\mathrm{km}}{R}\right) \qquad (2)

where: $G$ — Newton's gravitational constant; $c$ — speed of light. Reading: for a Schwarzschild black hole $GM/Rc^2=1/2$ ; a neutron star reaches about $40\%$ of that, so relativistic corrections to its structure are tens of percent. The structure is set by relativistic hydrostatic equilibrium — the outward pressure gradient against inward gravity in curved spacetime. This is the Tolman–Oppenheimer–Volkoff equation, the heart of the subject.

\frac{\mathrm{d}P}{\mathrm{d}r} = -\frac{G\,[\rho + P/c^2]\,[m(r) + 4\pi r^3 P/c^2]}{r^2\,[1 - 2Gm(r)/rc^2]} \qquad (3)

where: $P(r)$ — pressure at radius $r$ ; $\rho(r)$ — total mass–energy density; $m(r)$ — mass–energy enclosed within $r$ . Reading: every factor beyond the Newtonian $-Gm\rho/r^2$ makes gravity stronger — pressure gravitates ( $\rho\to\rho+P/c^2$ ), interior pressure adds to the source ( $m\to m+4\pi r^3P/c^2$ ), and the denominator encodes curvature — so all three push the maximum mass down. TOV is one equation for three functions, so it needs the mass-continuity relation that accumulates mass–energy shell by shell.

\frac{\mathrm{d}m}{\mathrm{d}r} = 4\pi r^2 \rho(r) \qquad (4)

where: $m(r)$ — enclosed mass–energy; $\rho$ — local mass–energy density. Reading: integrating outward from the centre tracks the gravitating mass inside radius $r$ ; the surface is where $P\to0$ , and $m(R)=M$ is the total mass. The system still has more unknowns than equations. What closes it — and what makes a neutron star a probe of nuclear physics — is the equation of state.

P = P(\rho) \quad \text{(equation of state)} \qquad (5)

where: $P(\rho)$ — microphysical pressure–density relation supplied by nuclear theory. Reading: TOV plus mass-continuity are two ODEs for three functions $P,\rho,m$ ; fixing $P(\rho)$ makes the system determinate, and each candidate EOS integrates to its own $M$ – $R$ curve — the single most important astrophysical unknown these stars probe. The simplest EOS is a cold, ideal degenerate neutron gas — the direct analogue of the white-dwarf electron gas.

P_{\rm deg} = \frac{\hbar^2}{15\pi^2 m_n}\left(3\pi^2 n\right)^{5/3} \quad (\text{non-rel. neutron gas}) \qquad (6)

where: $\hbar$ — reduced Planck constant; $m_n$ — neutron mass; $n$ — neutron number density. Reading: filling momentum states up to $p_F=\hbar(3\pi^2n)^{1/3}$ gives a $P\propto n^{5/3}$ polytrope; this ideal pressure is a lower bound on the truth, since at high density nuclear repulsion adds pressure the ideal gas lacks. How stiff can matter be? No perturbation can propagate faster than light, so the sound speed is bounded — a hard ceiling on stiffness.

c_s^2 = \frac{\mathrm{d}P}{\mathrm{d}\rho} \le c^2 \qquad (7)

where: $c_s$ — adiabatic sound speed. Reading: a stiffer EOS supports more mass, but $c_s\le c$ caps the stiffness, bounding the maximum mass at roughly $3\,M_\odot$ . Finally, the observable a merger delivers: the response of each star to its companion's tidal field, the dimensionless tidal deformability.

\Lambda = \frac{2}{3} k_2 \left(\frac{Rc^2}{GM}\right)^5 \qquad (8)

where: $\Lambda$ — dimensionless tidal deformability; $k_2$ — second (tidal) Love number, $\sim0.05\text{–}0.1$ . Reading: $\Lambda$ scales as the fifth power of $R/M$ , so it is exquisitely radius-sensitive; GW170817 measured $\Lambda_{1.4}\lesssim800$ , disfavouring the largest-radius equations of state.

Derivation AOrder-of-magnitude radius and maximum mass from degeneracy

The goal is to recover, from a single Newtonian energy balance, both the characteristic 10 km radius of a degeneracy-supported neutron star and — in the relativistic limit — the very existence of a maximum mass. This is the neutron analogue of the Chandrasekhar argument: it shows the maximum mass is not a nuclear accident but a robust consequence of quantum statistics fighting gravity. We treat the star as $N$ neutrons of mass $m_n$ in a ball of radius $R$ .

Step 1. Confine each neutron to volume $\sim R^3/N$ , so its momentum is $p_F\sim\hbar(N/R^3)^{1/3}$ and, non-relativistically, $E_F\sim p_F^2/2m_n\sim(\hbar^2/2m_n)(N/R^3)^{2/3}$ . Why: degeneracy pressure is just the kinetic energy the exclusion principle forces on fermions when you squeeze them.

Step 2. Add the competing kinetic and gravitational energies, using $M=Nm_n$ :

E(R) \sim N E_F - \frac{GM^2}{R} \sim \frac{\hbar^2 N^{5/3}}{m_n R^2} - \frac{G m_n^2 N^2}{R}

The kinetic term is positive ( $\propto R^{-2}$ ), the gravitational term negative ( $\propto R^{-1}$ ). Why: equilibrium is the radius that minimizes total energy, where these two trends cross.

Step 3. Set $\mathrm{d}E/\mathrm{d}R=0$ to find a genuine minimum at

R \sim \frac{\hbar^2}{G m_n^3 N^{1/3}} \sim 10\,\mathrm{km}\ \left(\frac{M}{1.4\,M_\odot}\right)^{-1/3}

For $N\sim1.7\times10^{57}$ this returns a few to ten kilometres — the right size. Why: kinetic energy falls faster with $R$ than gravity grows, so a stable minimum exists just as for a white dwarf, and the radius shrinks as mass grows.

Step 4. Now push the neutrons to relativistic speeds, $p_Fc>m_nc^2$ . The kinetic energy per particle becomes $\sim p_Fc$ , so $E\sim\hbar c\,N^{4/3}/R-Gm_n^2N^2/R$ . Both terms scale as $R^{-1}$ , and the sign of the total depends only on $N$ : above a critical $N_{\max}\sim(\hbar c/Gm_n^2)^{3/2}$ gravity wins at every radius. This yields $M_{\max}\sim(\hbar c/G)^{3/2}/m_n^2\sim M_\odot$ — the same combination as the Chandrasekhar mass. Why: the ultra-relativistic gas is a $\gamma=4/3$ polytrope, exactly marginally stable, so relativity converts "shrinking radius" into "hard mass ceiling". *This estimate reproduces $R\sim10\,\mathrm{km}$ for $N\sim10^{57}$ and $M_{\max}\sim$ a few $M_\odot$ ; the exact ideal-gas result in full GR is the Oppenheimer–Volkoff $0.7\,M_\odot$ .*

Go deeper · Derivation B · 1 minThe Newtonian limit of the TOV equation

Derivation BThe Newtonian limit of the TOV equation

The aim is to show how the TOV equation collapses to Newtonian hydrostatic equilibrium when relativity is switched off, isolating the three correction terms that make a neutron star qualitatively different from a white dwarf — which physics you may drop for a white dwarf and must keep for a neutron star. We start from the full equation and expand in the small parameters $P/\rho c^2$ and $2Gm/rc^2$ .

Step 1. Write the equation in full, exposing the three relativistic factors:

\frac{\mathrm{d}P}{\mathrm{d}r} = -\frac{G(\rho + P/c^2)(m + 4\pi r^3 P/c^2)}{r^2(1 - 2Gm/rc^2)}

Each parenthesis is a departure from Newton's $-Gm\rho/r^2$ . Why: naming the three factors separately lets us track which physical effect — pressure-as-source, pressure-as-mass, or curvature — each one carries.

Step 2. Impose the weak-field, low-pressure conditions $P/\rho c^2\ll1$ , $4\pi r^3P/mc^2\ll1$ , and $2Gm/rc^2\ll1$ . Why: for a white dwarf all three are tiny (parts in $10^3$ or smaller), so Newton is excellent; for a neutron star each is of order $0.2$ , so none may be dropped.

Step 3. Expand to leading order in these small quantities:

\frac{\mathrm{d}P}{\mathrm{d}r} \approx -\frac{G m \rho}{r^2}\left[1 + \frac{P}{\rho c^2} + \frac{4\pi r^3 P}{m c^2} + \frac{2Gm}{rc^2} + \cdots\right]

Dropping the bracket recovers Newtonian equilibrium $\mathrm{d}P/\mathrm{d}r=-Gm\rho/r^2$ ; keeping it steepens the pressure gradient. Why: every correction is positive, so GR always demands a steeper pressure gradient than Newton — the star must work harder to support itself, which is why $M_{\max}$ drops relative to a Newtonian estimate. *Each relativistic correction is $\sim20\%$ for a canonical neutron star.*

Go deeper · Derivation C · 1 minSurface gravitational redshift and apparent radius

Derivation CSurface gravitational redshift and apparent radius

The purpose is to turn the compactness into something a telescope can measure: the gravitational redshift of light climbing out of the star's potential well and the enlarged apparent radius that light-bending produces — the bridge from the abstract ratio $GM/Rc^2$ to spectral lines and thermal pulse profiles. We use the exterior Schwarzschild metric.

Step 1. A photon emitted at the surface radius $R$ and received at infinity is redshifted by the $g_{tt}$ component of the Schwarzschild metric:

\frac{\nu_\infty}{\nu_{\rm em}} = \sqrt{1 - \frac{2GM}{Rc^2}}

Why: clocks deep in a gravitational well run slow relative to infinity, so any periodic signal is dragged down.

Step 2. Invert to define the surface redshift $z$ :

1 + z = \frac{\nu_{\rm em}}{\nu_\infty} = \left(1 - \frac{2GM}{Rc^2}\right)^{-1/2}

For $M=1.4\,M_\odot$ and $R=10\,\mathrm{km}$ , $2GM/Rc^2\approx0.41$ , so $1+z\approx1.31$ , $z\approx0.31$ . Why: a measured line redshift reads out the single ratio $M/R$ directly — one number from one measurement, independent of any EOS assumption.

Step 3. The same curvature makes the star look bigger than it is; its apparent (radiation) radius seen at infinity is

R_\infty = R\left(1 - \frac{2GM}{Rc^2}\right)^{-1/2} = R(1+z)

Why: light-bending lets an observer see past the geometric limb onto part of the far hemisphere, inflating the inferred emitting area; thermal-emission fits exploit both the redshift and this bending to pin $M$ and $R$ jointly. *For canonical parameters, $2GM/Rc^2\approx0.41$ gives $z\approx0.31$ (not $0.35$ ); the apparent radius exceeds the coordinate radius by the factor $1+z$ .*

Limiting cases

Neutron-star physics interpolates between clean limiting regimes, each of which isolates one piece of the full problem:

Case	Result	Meaning
Ideal degenerate neutron gas (non-relativistic)	$P\propto\rho^{5/3}$ , $\ M_{\max}^{\rm OV}\approx0.7\,M_\odot$	Oppenheimer–Volkoff 1939: free-neutron gas in full GR. Too soft to match observed masses — proof nuclear forces are essential.
Ultra-relativistic degeneracy	$P\propto\rho^{4/3}$	When the Fermi energy exceeds $m_nc^2$ ; a $4/3$ polytrope is marginally unstable, forcing a maximum mass.
Newtonian limit	$\dfrac{\mathrm{d}P}{\mathrm{d}r}\to-\dfrac{Gm(r)\rho}{r^2}$	Recovered from TOV when $P\ll\rho c^2$ and $GM/Rc^2\ll1$ . Fine for white dwarfs, invalid for neutron stars.
Causal / maximally stiff EOS	$P=\rho c^2-\rho_s c^2$ , $\ c_s=c$	Demanding $c_s\le c$ caps the stiffness, bounding $M_{\max}\lesssim3\,M_\odot$ for plausible transition densities.

Common trapa neutron star is just a scaled-up white dwarf with neutrons instead of electrons

It is not, on two counts. First, an ideal degenerate neutron gas — the honest "neutrons for electrons" model — gives the Oppenheimer–Volkoff maximum mass of only $\sim0.7\,M_\odot$ , well below the $\sim2\,M_\odot$ pulsars we observe; the missing support comes from the repulsive core of the strong nuclear force, absent in the white-dwarf electron gas. Second, a white dwarf is Newtonian ( $GM/Rc^2\sim10^{-4}$ ) whereas a neutron star has $GM/Rc^2\sim0.2$ , so its structure needs the general-relativistic TOV equation. Treating a neutron star as a big white dwarf gets both the maximum mass and the mass–radius slope qualitatively wrong.

Worked examples

1 · The canonical neutron star by the numbers (fully worked)

Take the textbook fiducial star, $M=1.4\,M_\odot$ and $R=10\,\mathrm{km}$ . Find its mean density, its compactness, and its surface gravitational redshift, and say what each one tells you.

Step 1. Mean density from equation (1), with $M=2.78\times10^{33}\,\mathrm{g}$ and $R=10^6\,\mathrm{cm}$ :

\bar\rho = \frac{3M}{4\pi R^3} = \frac{3(2.78\times10^{33})}{4\pi(10^6)^3} = \mathbf{6.6\times10^{14}\ g\,cm^{-3}}.

That is $\sim2.5\,\rho_0$ — the average density exceeds a nucleus, so the interior is nuclear matter, not degenerate atoms.

Step 2. Compactness from equation (2). With $GM=1.86\times10^{26}$ and $Rc^2=10^6\times(2.998\times10^{10})^2=8.99\times10^{26}$ ,

\frac{GM}{Rc^2} = \frac{1.86\times10^{26}}{8.99\times10^{26}} = \mathbf{0.207}.

This is $\sim40\%$ of the black-hole value $1/2$ , so corrections are of order $20\%$ — general relativity is mandatory.

Step 3. Surface redshift from Derivation C, with $2GM/Rc^2=0.413$ :

1+z = (1-0.413)^{-1/2} = \mathbf{1.31},\qquad z=\mathbf{0.31}.

A surface line arrives at infinity redshifted by $31\%$ , and the apparent radius is $R_\infty=R(1+z)\approx13\,\mathrm{km}$ . Density $6.6\times10^{14}\,\mathrm{g\,cm^{-3}}$ , compactness $0.21$ , redshift $z\approx0.31$ — nuclear density, strong gravity, directly measurable.

2 · Are the neutrons relativistic?

At a representative interior density $\rho\approx6\times10^{14}\,\mathrm{g\,cm^{-3}}$ , estimate the neutron Fermi momentum and compare it to $m_nc^2$ . Does the non-relativistic formula (6) apply?

Step 1. Number density: $n=\rho/m_n=(6\times10^{14})/(1.675\times10^{-24})=\mathbf{3.6\times10^{38}\ cm^{-3}}$ .

Step 2. Fermi momentum $p_F=\hbar(3\pi^2n)^{1/3}$ : with $3\pi^2n=1.06\times10^{40}\,\mathrm{cm^{-3}}$ , the cube root is $2.2\times10^{13}\,\mathrm{cm^{-1}}$ , so $p_F=2.3\times10^{-14}\,\mathrm{g\,cm\,s^{-1}}$ and

p_Fc \approx 4.3\times10^{2}\ \mathrm{MeV}.

Step 3. Compare to the rest energy $m_nc^2=939\,\mathrm{MeV}$ : the relativity parameter is $x=p_F/m_nc=433/939\approx\mathbf{0.46}$ . The neutrons are mildly relativistic — Fermi kinetic energy $\approx95\,\mathrm{MeV}$ per particle. So formula (6) is only a first approximation; relativistic corrections are $\sim20\%$ , which with the nuclear interaction is why the realistic EOS departs from a $\rho^{5/3}$ polytrope. The interior sits right where both relativity and strong interactions matter — the regime that makes the EOS hard and interesting.

Physics of the portrait

The neutron star is where nuclear physics, quantum statistics, and general relativity were first forced to work together — and the person who did that, decades before anyone saw a pulsar, was J. Robert Oppenheimer.

Origins · 1 minJ. Robert Oppenheimer and how this was discovered

J. Robert Oppenheimer

1904–1967 · general-relativistic stellar structure and the neutron-star maximum mass

J. Robert Oppenheimer is remembered by the public for Los Alamos, but to astrophysicists his most enduring work is a pair of 1939 papers that founded the theory of compact objects. Neutron stars had been conjectured by Baade and Zwicky in 1934, just after Chadwick found the neutron — but no one had checked whether such a thing could exist in equilibrium. Working with George Volkoff and building on Richard Tolman's relativistic formalism, Oppenheimer wrote down what is now the Tolman–Oppenheimer–Volkoff equation and integrated it for an ideal degenerate neutron gas. The result was startling: an ideal-gas neutron star has a maximum mass of only about $0.7\,M_\odot$ , above which no equilibrium exists. In a companion paper with Hartland Snyder that same year, he showed that continued collapse produces what we now call a black hole — the first serious relativistic treatment of gravitational collapse. That low maximum mass, later understood as an artefact of neglecting nuclear forces, is exactly what makes the observed $2\,M_\odot$ pulsars so informative: they prove the true equation of state is far stiffer than the ideal gas he assumed, and the whole modern enterprise of constraining dense matter works out the tension he first exposed.

History

1932

— Chadwick discovers the neutron

1934

— Baade and Zwicky propose neutron stars as the remnants of supernovae

1939

— Oppenheimer and Volkoff integrate the TOV equation and find a maximum mass of about 0.7 M☉

1967

— Bell and Hewish detect the first radio pulsar, soon identified as a rotating neutron star

2017

— GW170817: the first neutron-star merger seen in gravitational waves and across the electromagnetic spectrum

Going further

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 16 (Compact Stellar Remnants) · Shapiro and Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (1983), ch. 2, 8–11 · Glendenning, Compact Stars, 2nd ed. (2000) · Oppenheimer and Volkoff, Phys. Rev. 55, 374 (1939) · Lattimer and Prakash, "The Physics of Neutron Stars", Science 304, 536 (2004). Key numerical anchors for this article: mean density $\bar\rho\approx6.6\times10^{14}\,\mathrm{g\,cm^{-3}}$ and compactness $GM/Rc^2\approx0.21$ for $M=1.4\,M_\odot$ , $R=10\,\mathrm{km}$ ; the Newtonian-limit expansion of TOV yields three $\mathcal{O}(0.2)$ corrections; the Oppenheimer–Volkoff ideal-gas maximum mass is $\approx0.7\,M_\odot$ ; the surface redshift is $z\approx0.31$ (not $0.35$ ) from $2GM/Rc^2\approx0.41$ ; the Fermi-momentum estimate gives $p_Fc\approx4.3\times10^2\,\mathrm{MeV}$ and $x\approx0.46$ at $\rho\approx6\times10^{14}\,\mathrm{g\,cm^{-3}}$ ; observed anchors are $M\approx2.08\,M_\odot$ (PSR J0740+6620), $R\approx12\,\mathrm{km}$ (NICER), and $\Lambda_{1.4}\lesssim800$ (GW170817). Nuclear saturation density $\rho_0\approx2.7\times10^{14}\,\mathrm{g\,cm^{-3}}$ is standard.

Prerequisites. Electron Degeneracy Pressure · Chandrasekhar Mass Limit · White Dwarfs · Tolman–Oppenheimer–Volkoff Equation. Leads to. Neutron Star Structure and Interior · Equation of State of Dense Matter · Pulsars · Magnetars · Millisecond and Recycled Pulsars · Pulsar Glitches and Neutron Star Superfluidity.

Ask the tutor: Want to integrate the TOV equation yourself for a toy polytrope and watch the mass–radius curve — and its maximum mass — appear? Ask and we'll build it step by step.

Why can't neutron degeneracy pressure alone support a 2 M☉ star, if it supports one at all?

Because an ideal degenerate neutron gas is too soft. When Oppenheimer and Volkoff integrated the TOV equation for free neutrons in 1939 they found a maximum mass of only about $0.7\,M_\odot$ — above that, no equilibrium exists. Real neutron stars exceed this because nuclear matter is far stiffer than an ideal gas: the strong interaction is strongly repulsive at short range, so squeezing nucleons together costs much more energy than the exclusion principle alone demands. That extra pressure raises the ceiling to $2\,M_\odot$ and beyond: degeneracy gets you a neutron star of sorts, but it takes the nuclear force to make one as heavy as those we see.

What physically happens at the maximum mass?

The maximum mass is the peak of the mass–radius curve and marks the onset of instability. Below it, adding mass makes the star shrink and its pressure rise enough to hold the extra weight — stable. At the peak, $\mathrm{d}M/\mathrm{d}\rho_c=0$ : adding mass no longer buys enough pressure, so compression runs away. The star collapses on a millisecond timescale, and because it is already so compact the collapse cannot halt — it forms a black hole. The maximum mass is thus simultaneously a statement about nuclear physics (how stiff can matter be?) and gravity (when is collapse inevitable?), which is why measuring it is prized.

Is a neutron star really "one giant nucleus"?

It is a useful and mostly honest picture, with two caveats. First, an ordinary nucleus is bound by the strong force and would fly apart under its own charge if large; a neutron star is bound by gravity, which lets it reach $10^{57}$ nucleons instead of a few hundred. Second, it is enormously neutron-rich — beta equilibrium under gravity drives the proton fraction to a few percent, whereas stable nuclei are roughly half protons. So: a giant, gravity-bound, extremely neutron-rich nucleus, its stiffness set by the same strong interaction.

Why does general relativity matter so much more for a neutron star than for a white dwarf?

It comes down to compactness, $GM/Rc^2$ . For a white dwarf this ratio is around $10^{-4}$ , so the three relativistic corrections in the TOV equation are parts in ten thousand — negligible, and Newton is superb. For a neutron star the ratio is about $0.2$ , so each correction is a $\sim20\%$ effect, and they all strengthen gravity, so a Newtonian calculation overestimates how much mass the star can hold. GR also has directly observable consequences here — a $31\%$ surface redshift, light-bending, and orbital decay in binary pulsars — none detectable for white dwarfs. The difference is one of magnitude, and the magnitude is decisive.

What is actually inside the core — do we know?

Not yet, and that is the point of studying these stars. The outer core is fairly settled: a neutron superfluid with a smaller admixture of superconducting protons, plus electrons and muons in beta equilibrium. The inner core, above roughly twice nuclear saturation, is genuinely uncertain — candidates include ordinary nucleonic matter, hyperons (heavier strange baryons), pion or kaon condensates, or deconfined quark matter, even a "hybrid" quark core inside a nucleonic mantle. Each softens or stiffens the EOS differently and predicts a different maximum mass and radius, which is why mass, radius, and tidal measurements are so valuable: they are our only probe of matter in a regime no laboratory can create.

How do astronomers actually measure a neutron star's radius?

Chiefly two ways, both relativistic. The first is X-ray pulse-profile modelling, the NICER method: a rotating star with hot polar caps produces an X-ray light curve whose shape depends on how gravity bends the emitted light — hence on the compactness $GM/Rc^2$ . Fitting the waveform returns $M$ and $R$ together, giving $R\approx12\,\mathrm{km}$ for the canonical mass. The second is tidal deformability from mergers: because $\Lambda\propto(R/M)^5$ , the tidal imprint on an inspiral like GW170817 is a sensitive radius probe. Older thermal-emission and burst methods work too, but a redshift-radius degeneracy makes them harder.